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August 31, 2014

Uncountably Categorical Theories

Posted by John Baez

Right now I’d love to understand something a logician at Oxford tried to explain to me over lunch a while back. His name is Boris Zilber. He’s studying what he informally calls ‘logically perfect’ theories — that is, lists of axioms that almost completely determine the structure they’re trying to describe. He thinks that we could understand physics better if we thought harder about these logically perfect theories:

His ways of thinking, rooted in model theory, are quite different from anything I’m used to. I feel a bit like Gollum here:

A zeroth approximation to Zilber’s notion of ‘logically perfect theory’ would be a theory in first-order logic that’s categorical, meaning all its models are isomorphic. In rough terms, such a theory gives a full description of the mathematical structure it’s talking about.

The theory of groups is not categorical, but we don’t mind that, since we all know there are lots of very different groups. Historically speaking, it was much more upsetting to discover that Peano’s axioms of arithmetic, when phrased in first-order logic, are not categorical. Indeed, Gödel’s first incompleteness theorem says there are many statements about natural numbers that can neither be proved nor disproved starting from Peano’s axioms. It follows that for any such statement we can find a model of the Peano axioms in which that statement holds, and also a model in which it does not. So while we may imagine the Peano axioms are talking about ‘the’ natural numbers, this is a false impression. There are many different ‘versions’ of the natural numbers, just as there are many different groups.

The situation is not so bad for the real numbers — at least if we are willing to think about them in a somewhat limited way. There’s a theory of a real closed field: a list of axioms governing the operations +,×,0+, \times, 0 and 11 and the relation ≤\le. Tarski showed this theory is complete. In other words, any sentence phrased in this language can either be proved or disproved starting from the axioms.

Nonetheless, the theory of real closed fields is not categorical: besides the real numbers, there are many ‘nonstandard’ models, such as fields of hyperreal numbers where there are numbers bigger than 11, 1+11+1, 1+1+11+1+1, 1+1+1+11+1+1+1 and so on.
These models are all elementarily equivalent: any sentence that holds in one holds in all the rest. That’s because the theory is complete. But these models are not all isomorphic: we can’t get a bijection between them that preserves +,×,0,1+, \times, 0, 1 and ≤\le.

Indeed, only finite-sized mathematical structures can be ‘nailed down’ up to isomorphism by theories in first-order logic. After all, the Löwenheim–Skolem theorem says that if a first-order theory in a countable language has an infinite model, it has at least one model of each infinite cardinality. So, if we’re trying to use this kind of theory to describe an infinitely big mathematical structure, the most we can hope for is that after we specify its cardinality, the axioms completely determine it.

And this actually happens sometimes. It happens for the complex numbers! Zilber believes this has something to do with why the complex numbers show up so much in physics. This sounds very implausible at first, but there are some amazing results in logic that one needs to learn before dismissing the idea out of hand.

Say κ\kappa is some cardinal. A first-order theory describing structure on a single set is called κ-categorical if it has a unique model of cardinality κ\kappa. And in 1965, a logician named Michael Morley showed that if a list of axioms is κ\kappa-categorical for some uncountable κ\kappa, it’s κ\kappa-categorical for every uncountable κ\kappa. I have no idea why this is true. But such theories are called uncountably categorical.

A great example is the theory of an algebraically closed field of characteristic zero.

When you think of algebraically closed fields of characteristic zero, the first example that comes to mind is the complex numbers. These have the cardinality of the continuum. But because this theory is uncountably categorical, there is exactly one algebraically closed field of characteristic zero of each uncountable cardinality… up to isomorphism.

This implies some interesting things. For example, we can take the complex numbers, throw in an extra element, and let it freely generate a bigger algebraically closed field. It’s ‘bigger’ in the sense that it contains the complex numbers as a proper subset, indeed a subfield. But since it has the same cardinality as the complex numbers, it’s isomorphic to the complex numbers!

And then, because this ‘bigger’ field is isomorphic to the complex numbers, we can turn this argument around. We can take the complex numbers, remove a lot of carefully chosen elements, and get a subfield that’s isomorphic to the complex numbers.

Or, if we like, we can take the complex numbers, adjoin a really huge set of extra elements, and let them freely generate an algebraically closed field of characteristic zero. The cardinality of this field can be as big as we want. It will be determined up to isomorphism by its cardinality. But it will be elementarily equivalent to the ordinary complex numbers! In other words, all the same sentences written in the language of +,×,0+, \times, 0 and 11 will hold. See why?

The theory of a real closed field is not uncountably categorical. This implies something really strange. Besides the ‘usual’ real numbers ℝ\mathbb{R} there’s another real closed field ℝ′\mathbb{R}', not isomorphic to ℝ\mathbb{R}, with the same cardinality. We can build the complex numbers ℂ\mathbb{C} using pairs of real numbers. We can use the same trick to build a field ℂ′\mathbb{C}' using pairs of guys in ℝ′\mathbb{R}'. But it’s easy to check that this funny field ℂ′\mathbb{C}' is algebraically closed and of characteristic zero. So, it’s isomorphic to ℂ\mathbb{C}.

In short, different ‘versions’ of the real numbers can give rise to the same version of the complex numbers! This is stuff they didn’t teach me in school.

There are purely mathematical arguments towards accepting the above for a definition of perfection. First, we note that the theory of the field of complex numbers (in fact any algebraically closed field) is uncountably categorical. So, the field of complex numbers is a perfect structure, and so are all objects of complex algebraic geometry by virtue of being definable in the field.

It is also remarkable that Morley’s theory of categoricity (and its extensions) exhibits strong regularities in models of categorical theories generally. First, the models have to be highly homogeneous, in a sense technically different from the one discussed for manifolds, but similar in spirit. Moreover, a notion of dimension (the Morley rank) is applicable to definable subsets in uncountably categorical structures, which gives one a strong sense of working with curves, surfaces and so on in this very abstract setting. A theorem of the present author states more precisely that an uncountably categorical structure MM is either reducible to a 2-dimensional “pseudo-plane” with at least a 2-dimensional family of curves on it (so is non-linear), or is reducible to a linear structure like an (infinite dimensional) vector space, or to a simpler structure like a GG-set for a discrete group GG. This led to a Trichotomy Conjecture, which specifies that the non-linear case is reducible to algebraically closed fields, effectively implying that MM in this case is an object of algebraic geometry over an algebraically closed field.

I don’t understand this, but I believe that in rough terms this would amount to getting ahold of algebraic geometry from purely ‘logical’ principles, not starting from ideas in algebra or geometry!

Ehud Hrushovski showed that the Trichotomy Conjecture is false. However, Zilber has bounced back with a new improved notion of logically perfect theory, namely a ‘Noetherian Zariski theory’. This sounds like something out of algebraic geometry, but it’s really a concept from logic that takes advantage of the eerie parallels between structures defined by uncountably categorical theories and algebraic geometry.

Models of Noetherian Zariski theories include not only structures from algebraic geometry, but also from noncommutative algebraic geometry, like quantum tori. So, Zilber is now trying to investigate the foundations of physics using ideas from model theory. It seems like a long hard project that’s just getting started.

Here’s a concrete conjecture that illustrates how people are hoping algebraic geometry will spring forth from purely logical principles:

The Algebraicity Conjecture. Suppose GG is a simple group whose theory (consisting of all sentences in first-order theory of groups that hold for this group) is uncountably categorical. Then G=𝔾(K)G = \mathbb{G}(K) for some simple algebraic group 𝔾\mathbb{G} and some algebraically closed field KK.

But there are many prerequisites I’m missing, and Richard Elwes, who studied with Zilber, has offered me some useful pointers:

If you want to really understand the Geometric Stability Theory referred to in your last two paragraphs, there’s a good (but hard!) book by that name by Anand Pillay. But you don’t need to go anywhere near that far to get a good idea of Morley’s Theorem and why the complex numbers are uncountably categorical. These notes look reasonable:

Basically the idea is that a theory is uncountably categorical if and only if two things hold: firstly there is a sensible notion of dimension (Morley rank) which can be assigned to every formula quantifying its complexity. In the example of the complex numbers Morley rank comes out to be pretty much the same thing as Zariski dimension. Secondly, there are no ‘Vaughtian pairs’ meaning, roughly, two bits of the structure whose size can vary independently. (Example: take the structure consisting of two disjoint non-interacting copies of the complex numbers. This is not uncountably categorical because you could set the two cardinalities independently.)

It is not too hard to see that the complex numbers have these two properties once you have the key fact of ‘quantifier elimination’, i.e. that any first order formula is equivalent to one with no quantifiers, meaning that all they can be are sets determined by the vanishing or non-vanishing of various polynomials. (Hence the connection to algebraic geometry.) In one dimension, basic facts about complex numbers tell us that every definable subset of ℂ\mathbb{C} must therefore be either finite or co-finite. This is the definition of a strongly minimal structure, which automatically implies both of the above properties without too much difficulty. So the complex numbers are not merely ‘perfect’ (though I’ve not heard this term before) but are the very best type of structure even among the uncountably categorical.

If you know anything else that could help me out, I’d love to hear it!

Posted at August 31, 2014 4:43 AM UTC

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Re: Uncountably Categorical Theories

So for any theory TT, we have a function gT:κ↦g_T: \kappa \mapsto the number of realizations of TT of size κ\kappa, where κ\kappa varies over cardinalities. I wonder if it’s reasonable to consider theories where gTg_T grows slowly, rather than actually being bounded by 11. (I don’t know how slowly, but gT(κ)≤κg_T(\kappa) \leq \kappa would be a start.)

Re: Uncountably Categorical Theories

@Allen, there’s a lot of deep work around that sort of question which goes by the name of “classification theory”, and my understanding is that the various possibilities are now well understood.

The key result was a theorem of Shelah that there is a dichotomy, in that either there are the maximum possible number of models 2^k in each uncountable cardinality k, or there are “few”. Subsequently others have analysed exactly what “few” means. There are more details on this Wiki page.

Re: Uncountably Categorical Theories

Shelah’s work “Classification Theory” can be considered one of the highlights of 20th century mathematics, although its influence in other branches of mathematics (and physics) has been more limited so far than advances in number theory or geometry for example. It is one of Shelah’s enormous contributions to mathematics which won him the Wolf prize in 2001.

Re: Uncountably Categorical Theories

Maybe not only the 20th century! Wilfrid Hodges wrote: “On any reckoning this is one of the major achievements of mathematical logic since Aristotle.”

I sometimes think of Zilber and other contemporary model theorists (or at least those working with the stability theory hierarchy) as cooking delicious meals using ingredients grown principally by Shelah.

(Also, for the record, I should make clear that I “studied with Zilber” in the sense of taking a 4th year undergraduate model theory course he taught at Oxford approx 15 years ago, while Jonathan Kirby was “a student of Zilber” in a stronger sense!)

Re: Uncountably Categorical Theories

The theory of a real closed field is not uncountably categorical.

One might as well mention an example that shows this. In fact Nonstandard Analysis is based on this possibility, that there are such real closed fields: one can take an ultrapower of the ordinary reals to get a bigger field ℝ*\mathbb{R}^\ast that contains infinitesimal elements (and which is therefore not isomorphic to ℝ\mathbb{R}).

In more detail: given an ultrafilter 𝒰\mathcal{U} on the natural numbers, an ultrapower is the set of functions f:ℕ→ℝf: \mathbb{N} \to \mathbb{R} (i.e., countable sequences of real numbers), modulo the equivalence relation where f∼gf \sim g if {n∈ℕ:f(n)=g(n)}\{n \in \mathbb{N}: f(n) = g(n)\} belongs to 𝒰\mathcal{U}. This ultrapower is again a real closed field. If 𝒰\mathcal{U} is not a principal ultrafilter, then the ultrapower ℝ*\mathbb{R}^\ast will definitely contain infinitesimal elements. On the other hand, it’s not hard to see that the cardinality of ℝ*\mathbb{R}^\ast is that of ℝ\mathbb{R}.

My mind boggles at nonprincipal ultrafilters: it seems at some intuitive level I refuse to think they exist. It’s a bit odd, because in some other contexts the axiom of choice seems perfectly fine to me. But nonprincipal ultafilters remind me too much of a thing that’s somewhere… but nowhere in particular. Sort of like that “damned elusive pimpernel” in The Scarlet Pimpernel:

Re: Uncountably Categorical Theories

No category theorist should for a moment consider “uncountable categoricity” to be any more “perfect” than the usual situation. Sure, every algebraically closed characteristic 00 field of cardinality 𝔠\mathfrak{c} is isomorphic to ℂ\mathbb{C}. But not in any canonical way. (This is due, more or less, to the hugeness of the Galois group of ℂ\mathbb{C} over ℚ\mathbb{Q}.) So I think it’s wrong to say that ℂ\mathbb{C} is “unique” in any reasonable way.

Indeed, I would say that the situation with ℝ\mathbb{R} is better: it has no field automorphisms. What’s more, pinning it down requires just deciding what “subset” means.

An orthogonal complaint is that the uncategorical categoricity of ℂ\mathbb{C} is not going to explain, without much further elaboration, the role of ℂ\mathbb{C} in quantum mechanics, for two important (although equivalent) reasons. Every attempt to provide a rigorous mathematical description of quantum mechanics I’ve ever seen really does use the reals: only “real” (i.e. self-adjoint) operators are considered “physical observables”; analytic questions of whether something is bounded, continuous, etc., really do matter when setting up even the axiomatics (not to mention posing all the interesting qualitative questions). A summary: the first order theory of a characteristic-00 algebraically closed field does not provide the language for Hilbert spaces.

Not that there isn’t deeply interesting mathematics here. But the claim that it will solve philosophical quesitons in physics seems to me dubious at best.

Re: Uncountably Categorical Theories

So I think it’s wrong to say that C is “unique” in any reasonable way.

I can’t agree with this statement! Most mathematicians would, I think, interpret “unique” as meaning “unique up to isomorphism”. Perhaps category theorists are atypical in this regard, I don’t know.

It’s perfectly true, as you say, properties of this type are essentially orthogonal to the property of “having lots of automorphisms”. (In fact one might even strengthen this to say the two are negatively correlated: the more automorphisms X has, the easier it may be for Y to be isomomorphic to X.)

Re: Uncountably Categorical Theories

There are good arguments for taking “unique” to mean “unique up to unique isomorphism” (in ordinary category theory) or more generally to mean “unique up to a contractible space of choices” (in higher category theory). As a simple example, it is dangerous to think of algebraic closures as unique: while they are unique up to isomorphism, they are not unique up to unique isomorphism, and you’ll get mixed up in arguments if you forget this (e.g. if you need to embed a field in its algebraic closure twice in an argument, you’ll get mixed up if you don’t keep track of whether you used the same embedding both times).

More generally, “unique up to a contractible space of choices” is equivalent to uniqueness in a very strong sense, namely in arbitrary families over an arbitrary base space. For example, vector spaces of a fixed dimension are unique up to isomorphism, but there are lots of interesting vector bundles over interesting base spaces. This is, to me, a very strong argument for refusing to refer to “the vector space of dimension n,” for example, in the same way that I don’t refer to “the algebraic closure.”

Re: Uncountably Categorical Theories

Re: Uncountably Categorical Theories

Theo wrote:

No category theorist should for a moment consider “uncountable categoricity” to be any more “perfect” than the usual situation. Sure, every algebraically closed characteristic 00 field of cardinality 𝔠\mathfrak{c} is isomorphic to ℂ\mathbb{C}. But not in any canonical way. (This is due, more or less, to the hugeness of the Galois group of ℂ\mathbb{C} over ℚ\mathbb{Q}.) So I think it’s wrong to say that ℂ\mathbb{C} is “unique” in any reasonable way.

Personally, instead of using morally loaded words like “perfect” or “reasonable” here, I prefer to just state the facts: all algebraically closed characteristic 00 field of a fixed uncountable cardinality are isomorphic, but not in a unique way. So, the groupoid of these fields is connected, but not contractible. Thinking of it as a homotopy type, its π0\pi_0 is trivial, but not its π1\pi_1.

(I mentioned “perfection” not because I think that way, but because I was trying to explain Zilber’s thought.)

An orthogonal complaint is that the uncategorical categoricity of ℂ\mathbb{C} is not going to explain, without much further elaboration, the role of ℂ\mathbb{C} in quantum mechanics…

I agree with that. Physicists use the standard topology on ℂ\mathbb{C} all over the place (like in electrical engineering), and quantum mechanics seems to rely fundamentally on its *\ast-algebra structure. So, when it comes to physics, I think of Zilber’s work as a tentative first exploration of something that would require fundamentally new ideas to succeed. There’s certainly no obvious reason why being uncountably categorical is either a necessary or sufficient reason for a mathematical structure to be important in physics.

It might turn out that model theory has nothing interesting to say about mathematical physics. After all, in physics we’re often more interested in syntax, i.e., calculations starting from axioms. Since when do physicists really care about set-theoretic models of the axioms? For example: who among them really cares if the real numbers include infinitesimals or not? Newton used them; nothing bad happened when people stopped using them; nothing bad would happen if we started again. It doesn’t seem to affect any experimental predictions.

Or, it could be that model theory has interesting things to say about mathematical physics, but we just haven’t figured out what. This is what I’m actually hoping. It could easily take a century to get to the bottom of this. Maybe uncountable categoricity will turn out to be less important than some other logical properties of theories.

When it comes to algebraic geometry rather than physics, the model theorists seem to be on much firmer ground. Hrushovski has already used model theoretic techniques to prove the geometric Mordell–Lang conjecture in all characteristics. Algebraic geometry more often avoids working with the usual topology and *\ast-algebra structure on ℂ\mathbb{C}. So maybe the slogan “uncountably categorical ≈\approx logically perfect” is suited to algebraic geometry, not physics.

Re: Uncountably Categorical Theories

Re: Uncountably Categorical Theories

My objection to this line of reasoning is that it doesn’t say anything about the usual topology on C, which seems important to me in applications to physics. Actually there’s a much older theorem (of Pontryagin?) about uniqueness statements in this setting: IIRC the uniqueness theorem is that R and C are the unique (up to isomorphism) connected locally compact fields, or something like that.

Re: Uncountably Categorical Theories

Re: Uncountably Categorical Theories

Good question! I have no idea. Regarding ℂ\mathbb{C}, the key result seems to be Steinitz’s Theorem saying that the isomorphism type of an algebraically closed field is determined by its characteristic and transcendence degree… plus the fact that for uncountable algebraically closed fields the transcendence degree is the same as the cardinality. I don’t know if people have analyzed the use of choice in these results.

Somewhat separately, one could ask if Morley’s categoricity theorem relies on the axiom of choice.

Re: Uncountably Categorical Theories

The uncountable categoricity of the complex field does depend on the axiom of choice, as does Morley’s Theorem. However, there is a sense in which that is slightly missing the point.

Morley’s theorem states that a first-order theory T which is categorical in one uncountable cardinality is uncountable in all uncountable cardinalities. However, the proof is even more interesting than the result. The proof explains that there is a strong structure theory for models of T. In a model M, every infinite definable set has a dimension (like the dimension of a vector space or the transcendence degree of a field). Furthermore, every definable set in a given model M has the same dimension. Why? Well, essentially because any two definable sets A and B must be in definable finite-to-finite correspondence with each other. That is not quite true, better to say that there are natural numbers nin_i such that An1A^{n_1} is in finite-to-finite correspondence with Bn2B^{n_2}. But from these correspondences one essentially shows that every definable set can be coordinatized in terms of some fixed definable set. For example, any complex algebraic variety can have complex coordinates put on it, so it is in correspondence with some affine space (at least on a open subset of each irreducible component, and then the singular set is of lower dimension and you can repeat the process). This is very roughly a way of thinking about the structure theory for definable sets in the first-order theory of algebraically closed fields.

Now, while the categoricity theorem depends on the axiom of choice, this structure theory for definable sets does not. Any while you may be forgiven for thinking that model theorists care mainly about models of theories, often that is just as a means to understanding definable sets, which are really just a manifestation of syntactic formulas.

Re: Uncountably Categorical Theories

Thanks for the expert assistance, Jonathan! It makes me more eager to understand the proof of Morley’s categoricity theorem.

This confuses me:

Furthermore, every definable set in a given model MM has the same dimension.

I thought this ‘dimension’ behaved like the dimension of a vector subspace or algebraic variety. For example, I thought that if we started with the theory of an algebraically complete field of characteristic zero (say ℂ\mathbb{C}), then the set {z=0}\{z = 0 \} would have dimension 0 while the whole field would have dimension 1.

Re: Uncountably Categorical Theories

Yes, in attempting to give a quick outline of the ideas I have glossed over something important.

Really a definable set S has a natural number dimension, so e.g. in the complex field, an algebraic variety’s dimension is what an algebraic geometer thinks it should be. Furthermore, a model M (ie an algebraically closed field of characteristic 0 in this example) also has a dimension, which in this case is its absolute transcendence degree (which is a cardinal). These two dimension notions are different, but closely related.

Now basically definable sets in this theory ACF0ACF_0 are varieties, and their realisations in a model M are just the M-points of those varieties. If you look at the M points of any two varieties (of positive dimension), say V(M) and U(M), you can get at the transcendence degree of M from both, and it will be the same cardinal. Here’s a way to get td(M) from V(M). Say for example td(M) is 3, and V is just the affine line. Then in V(M) we can find 3 points (a,b,c) which are “independent” from each other, in that they lie in V3V^3 but not in any proper subvariety. However, we can’t find 4 such points. The same will hold if V is any algebraic curve. However if U is the affine plane, the same calculation doesn’t work so well - we have to take account of the fact that U is 2-dimensional as a definable set. What does this mean? Well, essentially it means the points in U(M’), (for any model M’) are in finite-to-finite (in this case 1-1) correspondence with pairs of points in a 1-dimensional set, in this case V(M’).

It is not obvious how to do all of this abstractly without referring to fields, varieties and pre-existing notions of dimensions. The exciting thing is that it can be done in great generality.

Re: Uncountably Categorical Theories

Ah, excellent! As usual, a result that seems to depend on AC can be rephrased (more) constructively so as to be independent of it. That makes me much happier, and much more willing to regard this as some sort of “perfectness” property.

Re: Uncountably Categorical Theories

I have not had the time to read up on this.
Does anyone have a clear picture of the connection with topos theory? There is a bit in the Elephant and there is quite some work (Coste, Lombardi, Roy, …) on RCFs using topos theoretic methods.

Re: Uncountably Categorical Theories

I think that the real kick in the teeth about real-closed fields, is that while this theory is not categorical in any cardinality, if you add just one algebraic number to a real-closed field (namely, the square root of -1) you get an algebraically closed field of the same cardinality.

So while there are many different real-closed fields of size continuum, if you add them just this one element… it’s all collapsing down the the complex numbers. Bam!

Re: Uncountably Categorical Theories

Yes, that’s indeed a “kick in the teeth”. I mentioned it in my post:

The theory of a real closed field is not uncountably categorical. This implies something really strange. Besides the ‘usual’ real numbers ℝ\mathbb{R} there’s another real closed field ℝ′\mathbb{R}', not isomorphic to ℝ\mathbb{R}, with the same cardinality. We can build the complex numbers ℂ\mathbb{C} using pairs of real numbers. We can use the same trick to build a field ℂ′\mathbb{C}' using pairs of guys in ℝ′\mathbb{R}'. But it’s easy to check that this funny field ℂ′\mathbb{C}' is algebraically closed and of characteristic zero. So, it’s isomorphic to ℂ\mathbb{C}.

In short, different ‘versions’ of the real numbers can give rise to the same version of the complex numbers! This is stuff they didn’t teach me in school.

Re: Uncountably Categorical Theories

In about 1997, I sat in on a model theory course given by Martin Hyland. The very first thing he explained, as “motivation” for the rest of the course, was what he described as a logician’s proof of the Nullstellensatz.

I can write out this (short!) proof if you want; I think you’ll like it, if you don’t know it already. But the aspect I particularly wanted to share was the following. At some point in the proof, Martin invoked Hilbert’s basis theorem. At some later point, he invoked the fact that any two algebraically closed fields of the same characteristic and the same uncountable cardinality are isomorphic — and made the comment that this fact was “morally trivial compared to Hilbert’s basis theorem”.

Re: Uncountably Categorical Theories

John, I guess your first interpretation of Martin’s “morally trivial” is closer to what he intended. I took it to mean that the Hilbert basis theorem is deep, but this set-theoretic fact about fields isn’t (though myself, I don’t even know a proof). I should just ask him.

Here goes with the logician’s Nullstellensatz. By the looks of my notes, Martin did a lot of it out loud rather than on the board, so there is a distinct possibility of mistakes.

We’ll prove: for any proper ideal II of ℂ[x1,…,xn]\mathbb{C}[x_1, \ldots, x_n], the set
V(I)={a∈ℂn:f(a)=0 for all f∈I}
V(I) = \{ a \in \mathbb{C}^n : f(a) = 0 \,\,\text{ for all }\,\, f \in I\}
has at least one point.

Take a maximal ideal mm containing II. Then ϕ\phi holds in the field R=ℂ[x1,…,xn]/mR = \mathbb{C}[x_1, \ldots, x_n]/m, and therefore also holds in the algebraic closure R¯\bar{R} of RR (which is an extension of ℂ\mathbb{C}).

(So, we want to show that if some existential statement ϕ\phi holds in an extension of ℂ\mathbb{C}, then it holds in ℂ\mathbb{C} itself. That is, we want ℂ\mathbb{C} to be “existentially closed” — “algebraically closed ⇒\implies existentially closed”.)

[Here my notes look a bit unreliable, but I think the point is as follows.] Now consider ideals generated by a fixed number kk of polynomials of a fixed maximal degree dd, and let ψ\psi be the proposition:
for all polynomials f1,…,fk of degree at most d.∃x.(f1,…,fk) is a proper ideal ⇒f1(x)=⋯=fk(x)=0.
\begin{aligned}
\text{for all polynomials }\,\, f_1, \ldots, f_k
\,\,\text{ of degree at most }\,\, d.
\exists \mathbf{x}.\\
(f_1, \ldots, f_k) \,\,\text{ is a proper ideal }
\implies
f_1(\mathbf{x}) = \cdots = f_k(\mathbf{x}) = 0.
\end{aligned}
(Martin remarked that the part of that proposition strictly after “∃x\exists \mathbf{x}” is “actually quantifier-free”, but I don’t see why that’s true and it doesn’t seem to be needed in what follows.)

Set-theoretic thought: any two algebraically closed fields of a given characteristic and of the same uncountable cardinality are isomorphic.

But ψ\psi is true in one algebraically closed field of continuum cardinality and characteristic zero (namely, R¯\bar{R}), so it’s true in ℂ\mathbb{C} too.

Re: Uncountably Categorical Theories

There is something I don’t understand here. (Note: I’m not a mathematician.) You say that Complex Numbers (C) is “perfect” but its subset Natural Numbers (N) is not. Mustn’t this mean you forgot to include some important axioms of N when listing what you considered to be the important axioms of C? Otherwise, take any undecidable statement about N, and it’s also undecidable about C, isn’t it? Since it’s not hard to define N within C as the smallest closure under addition of {0,1}, unless you “forgot” to include standard set theory (for defining “smallest” or “closure”) in your toolkit of axioms about C. But if you did, then this quality of “perfection” is not so much about C vs. N as about which set of axioms you feel comfortable using at the time.

Re: Uncountably Categorical Theories

With luck a logician can give you more details. But one intuition is that ℂ\mathbb{C} doesn’t “know” whether a number is or isn’t in ℕ\mathbb{N}. The language is different.

Here’s a manifestation: you say “ℕ\mathbb{N} is the additive closure of {0,1}\{0,1\}”, which I will rewrite as “a number c∈ℂc\in \mathbb{C} is in ℕ\mathbb{N} if it is a sum of some number of 11s”. But this isn’t a 1st-order sentence. How many 11s? A natural number of them … but that’s what we’re trying to define. Actually, you were more precise: you said “the smallest closure …”. But to define that, you need to quantify over subsets of ℂ\mathbb{C} with some property, and that’s the very essence of second-order mathematics.

Re: Uncountably Categorical Theories

But if you did, then this quality of “perfection” is not so much about ℂ\mathbb{C} vs. ℕ\mathbb{N} as about which set of axioms you feel comfortable using at the time.

Right.

For ℕ\mathbb{N}, I said that the Peano axioms are incomplete; it’s also undecidable if a statement follows from those axioms.

For ℂ\mathbb{C}, I said that the axioms for an algebraically closed field of characteristic zero are complete; they’re also decidable. These axioms are merely:

• the ‘field’ axioms: the usual algebraic identities involving +,×,0+, \times, 0 and 11 together with laws saying that anything has an additive inverse and anything except 0 has a multiplicative inverse.

Note we have nothing like mathematical induction or the concept of ‘subset of complex numbers’. We can’t define exponentiation, so we can’t define π\pi, much less prove that it exists.

Indeed, if we restrict attention to algebraic complex numbers (roots of polynomials with integer coefficient), we get a perfectly fine countable model of this theory. We’d get another, nonisomorphic countable model if we took the smallest algebraically closed field containing the algebraic complex numbers together with one transcendental number, like π\pi. Or, we could throw in both π\pi and ee, which are both transcendental and ‘independent’: they don’t satisfy any nontrivial polynomial relations with each other.

So, there are tons of different countable models. This makes Steinitz’s Theorem seem rather wonderful: up to isomorphism, there’s just one model of each uncountable cardinality.

But in fact, maybe it’s not so wonderful. I believe all these models are just the algebraic complex numbers with some uncountable number of independent transcendentals thrown in.

What makes the complex numbers more interesting than ‘algebraic numbers with an enormous wad of independent extra numbers thrown in’ is their topology… or equivalently, complex conjugation (which allows us to define the distance function |x−y||x - y|).

Zilber’s approach to treating the complex numbers as merely an algebraically closed field of characteristic zero is quite common in algebraic geometry (which actually spends a lot of time exploring the relation between this attitude and the attitude where you also study the topology).

You might think the incompleteness and undecidability of the natural numbers are due to including the principle of mathematical induction in the Peano axioms. But Robinson arithmetic leaves out induction and is still incomplete and undecidable! So it’s a bit subtle.

Re: Uncountably Categorical Theories

… up to isomorphism, there’s just one model of each uncountable cardinality.

So throwing in pi, vs. throwing in both pi and e, you get “different” but isomorphic models? (I presume both models are countable.)

Also, about your earlier example of getting “the same C” whether forming it from pairs (R, R) or (R’, R’), is the same sort of thing going on, where you are also forgetting some of what we usually think of as important axioms of R? Or is it just no longer being able to find R (or R’) in the resulting C, sort of like Theo’s explanation of not being able to find N inside C? (Or both?)

About relevance to physics – my initial hope after reading your introduction was that the relevance was obvious in this sense: a physical theory has to make predictions; if it’s “perfect” then these predictions (in principle) include everything important (enough to tie down the model up to isomorphism).

But seeing that this is only achieved when importance aspects like topology (of sets of numbers) are being left out, I’m doubtful that an argument like that still makes sense.

Here’s a hand-wavier version of my initial hoped-for argument: if Nature runs the world according to some theory, then since Nature has to always be able to come up with successor states, it better always be able to uniquely-enough solve its time evolution equations. If they have non-isomorphic models this might be harder.

That is so vague that I’m having trouble figuring out if it gets even worse if applied to theories without any concept of topology.

BTW is there anything else important being left out, besides topology?

Re: Uncountably Categorical Theories

Bruce wrote:

So throwing in π\pi, vs. throwing in both π\pi and ee, you get “different” but isomorphic models?

No, when I said they’re “different” I really meant they’re not isomorphic! It’s too easy to get models that are unequal but still isomorphic: just name things differently. For example, throwing in π\pi versus throwing in ee. In the language where all you have is +,×,0,1+, \times, 0, 1, these give isomorphic models! There is no way to tell the difference between π\pi and ee in this restricted language.

(Here I’m using ‘models’ to mean algebraically closed fields of characteristic zero, since right now that’s the theory whose models we’re talking about.)

(I presume both models are countable.)

Yes, I was trying to illustrate how we can get lots of nonisomorphic countable models.

The model obtained by throwing in both π\pi and ee has “transcendence degree 2”. The transcendence degree is the maximum size of a set of algebraically independent elements, meaning ones that don’t obey any nontrivial polynomial equations (here with coefficients in the algebraic numbers, which is our starting-point for throwing in further numbers).

Models with different transcendence degree can’t be isomorphic, and it’s pretty easy to get models with any finite transcendence degree. All these are countable.

Also, about your earlier example of getting “the same ℂ\mathbb{C}” whether forming it from pairs (R, R) or (R’, R’), is the same sort of thing going on, where you are also forgetting some of what we usually think of as important axioms of ℝ\mathbb{R}?

I’m again taking a “purely algebraic” approach to the real numbers, leaving out the all-important axiom schema that there’s no number bigger than 1,1+1,1+1+1,…1, 1+1, 1+1+1, \dots and also the axiom that every bounded subset has a least upper bound. The latter axiom can’t even be formulated in first-order logic unless we introduce axioms for set theory!

In this “purely algebraic” approach we treat the real numbers as a real closed field. So, the axioms here are:

the ‘field’ axioms: the usual algebraic identities involving +,×,0+, \times, 0 and 11 together with laws saying that anything has an additive inverse and anything except 00 has a multiplicative inverse.

the ‘formally real field’ axiom, saying that −1-1 is not the square of anything. This implies that we can equip the field with a reasonable concept of ≤\le, but not necessarily in a unique way.

the ‘real closed’ axiom, which says that also for number xx either xx or −x-x has a square root, and every polynomial of odd degree has a root. Among other things this gives a unique reasonable concept of ≤\le: x≤yx \le y if y−xy - x has a square root.

The ‘real closed’ axiom is a watered-down version of the ‘algebraically closed’ axiom I mentioned in my last comment. The complex numbers are algebraically closed, but the real numbers are just real closed.

Like the axioms for an algebraically closed field, the axioms for a real closed field are complete and decidable. But the latter axioms have zillions of nonisomorphic models of any given uncountable cardinality, while the former have just one.

If you want to ‘blame’ this fact on something, maybe you can blame it on the fact that the axioms for a real closed field let us define ≤\le.

This lets us distinguish between what we get when we take the algebraic real numbers and throw in π\pi, and what we get when we throw in ee… since one is bigger than 3 and the other is smaller! More to the point, it lets us take the real numbers and get another uncountable model where we throw in a number that’s bigger than all the usual reals.

Btw is there anything else important being left out, besides topology?

In the complex case we’re leaving out complex conjugation and thus the concept of distance, |x|=xx¯|x| = \sqrt{x\overline{x}}.

In the real case we have the concept of distance, |x|=x2|x| = \sqrt{x^2}, and the concept of ≤\le, so we do have some prerequisites for topology. We can say that π\pi is between 3.14 and 3.15, which we couldn’t do in the complex case.

But we’re not letting ourselves talk about sets: our logical resources are just ∀,∃,∨,∧,¬,=\forall, \exists, \vee, \wedge, \not, = and variables. So we can’t say the most interesting thing about the reals: every bounded set has a least upper bound. So, we can’t exclude models consisting of just algebraic real numbers.

Furthermore, we haven’t bothered to insert the axiom schema that rules out ‘infinitely big’ numbers: numbers greater than 1,1+1,1+1+1,…1, 1 + 1, 1+ 1+ 1, \dots. This means we get interesting models with infinities and infinitesimals. Since the theory of real closed fields is complete, these other ‘weird’ models make the exact same sentences true (in our restricted language).

And as Richard Elwes once pointed out to me, Schanuel’s conjecture is deeply related to the model theory we’re talking about here. An exponential field is field with an operation exp\exp obeying the usual algebraic rules of exponentiation

Zilber showed there is a unique exponential field that obeys Schanuel’s conjecture and has the ordinary complex numbers as its underlying field!

So either the complex numbers with its usual notion of exponentiation obeys Schanuel’s conjecture, or there is a ‘pseudoexponentiation’ on the complex numbers which obeys exp(a+b)=exp(a)exp(b)\exp(a + b) = \exp(a) \exp(b), exp(0)=1\exp(0) = 1, obeys Schanuel’s conjecture, but is not the usual exponentiation!

This is a very interesting ‘rhetorical argument’ for Schanuel’s conjecture: either it’s true or something very weird would be true.

Re: Uncountably Categorical Theories

Re: Uncountably Categorical Theories

More about relevance to physics: there are comments far above, from both Theo and John Baez, which are skeptical that this “perfect” quality is relevant to physics. And here is another one (which might be related to those): I don’t know how to use any physical theory to make predictions, unless it can calculate probabilities of observations, which seems to imply it “understands real numbers” (including their topology, or at least enough of it to talk about whether some probability is in some interval). I also don’t know how to use it unless I am allowed to express my initial conditions as a probability distribution, or at least, to require them to lie in some “small” subset of configuration space – that is, it’s important that I’m not required to specify the initial conditions exactly, to make a prediction.

So does anyone here understand an argument on the other side, i.e. a specific reason anyone suspects “perfection” per se might be relevant to physics?

Perhaps related – is it ruled out that any “perfect” theory can include topology, reals, etc, or merely not yet known whether this is possible?

Re: Uncountably Categorical Theories

Perhaps related — is it ruled out that any “perfect” theory can include topology, reals, etc, or merely not yet known whether this is possible?

There’s a lot known about this kind of stuff, but I don’t know most of it. With his Trichotomy Conjecture and subsequent refinements, Zilber and others are working toward a kind of ‘classification’ of uncountably categorical structures. Very very roughly, the Trichotomy Structure and its corrected versions say all ‘logically perfect’ theories involve either 1) a linear order, 2) a vector space over a division ring, or 3) an algebraically closed field.

If we want to get ordinary physics into the game, I think we need something like the real numbers seen not just as a real closed field but as a complete ordered field. There are some standard axioms for an ordered field. If we use second-order logic (which is rather unpopular) we can quantify over predicates and add an extra axiom saying “for all predicates PP, if for some xx we have P(y)P(y) only for y≤xy \le x, then there is a leastxx such that P(y)P(y) holds only for y≤xy \le x.”

The resulting theory is categorical: it has only one model! So, in some sense this is even more “perfect” than the uncountably categorical theories we’ve been discussing. But many logicians dislike second-order logic because there’s not such a nice match of syntax and semantics as in first-order logic. Unlike first-order logic, second-order logic has no system of deduction rules that’s:

Sound: Every provable second-order sentence holds in every model (using the standard concept of model).

Complete: Every formula that holds in every model is provable.

Effective: There is an algorithm that can correctly decide whether any given sequence of symbols is a valid proof.

So, most logicians consider second-order logic too defective to be happy that — unlike first-order logic — it admits theories that have a unique (up to isomorphism) infinite model.

Re: Uncountably Categorical Theories

So, most logicians consider second-order logic too defective …

What if you try to avoid that by introducing, not a 2nd order axiom over predicates P, but a first order axiom schema over formulas P (perhaps with additional parameters, for which the axiom uses additional quantifiers)?

(I am not sure whether that could capture what you actually want to do to use the original 2nd order axiom, when formalizing some physical theory.)

From what you said earlier I presume that this doesn’t (or at least is not known to) lead to a “logically perfect” theory which includes the reals.

Re: Uncountably Categorical Theories

Bruce wrote:

What if you try to avoid that by introducing, not a 2nd order axiom over predicates P, but a first order axiom schema over formulas P (perhaps with additional parameters, for which the axiom uses additional quantifiers)?

I don’t know. I don’t even know if people have studied that theory!

But here’s something it intensely reminds me of.

The original axioms of Peano arithmetic, due to Peano himself, were second-order. He phrased mathematical induction by quantifying over predicates: “for all PP, if P(0)P(0) and ∀n(P(n)⇒P(n+1))\forall n (P(n) \Rightarrow P(n+1)), then for ∀nP(n)\forall n P(n).” Dedekind showed in 1888 that any two models of this theory are isomorphic.

However, second-order logic is so annoying that people reformulated Peano arithmetic in a first-order way using an axiom schema over formulas PP. And then we get lots of nonisomorphic models.

Of course we can try to get around this by embedding arithmetic in a first-order formulation of set theory and quantifying over subsets S⊆ℕS \subseteq \mathbb{N}. Gödel’s theorems still apply: now we get lots of models of set theory. But maybe “up to that” we get a “relatively unique” model of the natural numbers. I don’t know, because I don’t know how to make this idea precise. But somehow it’s hard to imagine the situation here is substantially worse than in the original second-order formulation… especially since the concept of “model” used in the second-order formulation relies on set theory!

Re: Uncountably Categorical Theories

I think in the real-ordered-field setting there just aren’t enough formulas; all you can mention is inequalities of polynomials, and the ROF axioms already say the extremal cases (replace all ≤\leq with ==) have as many solutions as possible. There’s no way to get analysis into it.

Re: Uncountably Categorical Theories

Just a pointer, there is a notion of an ω\omega-model of set theory, namely one where the natural numbers are the ‘standard’ natural numbers, or rather, that there are no non-standard natural numbers. This is relative to some agreed-upon meta theory; internally every model ‘believes’ every natural number therein is standard.

Re: Uncountably Categorical Theories

I think that’s another good reason to consider second-order logic defective, at least from a foundational viewpoint. It’s not just that logicians have a fetish about some properties that it fails to satisfy; it’s that the very concept of “second-order logic” implicitly assumes a background set theory. So the idea that in second-order logic you can “uniquely” specify a structure is misleading, because you’ve only specified it uniquely relative to your background set theory, which is itself in no way unique.

Re: Uncountably Categorical Theories

It’s not just that logicians have a fetish about some properties that it fails to satisfy; it’s that the very concept of “second-order logic” implicitly assumes a background set theory

I’m not sure I understand this: is the idea that the very concept of second-order logic implicitly assumes set theory because second-order variables intuitively correspond to collections of things, whereas first-order variables intuitively correspond to things? It seems to me too strong to say that the only intuitively correct formal way to interpret predicates is as subsets: it is conceivable that we could develop a theory of predicates without a background set theory. Indeed you could use second-order logic as the metatheory of second-order logic and that would certainly not assume a background set theory. But perhaps that is your point? The fact that second-order logic can be its own metatheory shows that it can handle a notion of satisfaction, and anything that can handle a notion of satisfaction has some kind of set theory (naive or otherwise) implicit in it?

Re: Uncountably Categorical Theories

I think what Mike is getting at is that second-order logic won’t actually fit in a foundational theory: it’s a defined thing within your-favourite-foundations. In that sense, the details of the second-order theory “complete ordered field” tells us more about the ambient set theory than it tells us about the usual ordering of the rationals; Cf. the continuum hypothesis.

Saying something slightly different, the theory “two complete ordered fields in a model of ZF” is a first-order theory, and one of the theorems of that theory is a canonical isomorphism of those two complete ordered fields.

Re: Uncountably Categorical Theories

Right, I completely agree that usually when we talk about second-order logic we define it within some fixed foundation, e.g. ZFC. As such the metatheory of second-order logic is usually some kind of set theory and, as Mike points out, the above-quoted categoricity results are not really inherent properties of second-order logic but rather of the metatheory in which we are proving them, e.g. ZFC. I wasn’t questioning this.

What I was skeptical about was the (implicit) assertion that it has to be this way: why are we a priori restricted to a set-theoretic metatheory for second-order logic? Nothing, it seems to me, would prevent me from using second-order logic as my foundation (perhaps with extra axioms to make it resemble some set theory) and therefore also as the metatheory of the (internal) second-order logic I define within this foundation. In such a situation, it seems to me, second-order logic does not require a previously fixed background set theory - although, of course, it requires a background second-order logic that, one may argue, is just as bad. I’m not sure what categoricity results for (internal) second-order logic would look like in this set-up though.

Re: Uncountably Categorical Theories

Mike wrote:

it’s that the very concept of “second-order logic” implicitly assumes a background set theory.

Like Dimitris, I’m uncomfortable with this formulation of the problem. We can choose deduction rules for second-order logic and use them without having previously chosen some other axioms for some set theory that chugs away in the ‘background’.

(We do need a background set theory to develop the model theory of second-order logic, but we also need it to develop the model theory of first-order logic.)

I prefer this formulation:

I’m saying that a “theory of predicates” essentially is a background set theory of a sort.

I’d just say I don’t think there’s anything ‘background-y’ about it: it’s right there in the foreground, described using axioms and deduction rules.

Admittedly, the deduction rules for 2nd-order logic seem a bit problematic, and I don’t understand them well. When people try to keep improving them, it seems they get systems that more and more closely resemble set theory formulated in first-order logic. So perhaps the whole idea is a bit of a will-o’-the-wisp.

Re: Uncountably Categorical Theories

Well, it’s in the background if you’re talking about semantics of second-order logic, as you are when you say things like that second-order logic enables you to characterize certain structures uniquely.

But really I think we’re all just saying the same thing in different ways.

Re: Uncountably Categorical Theories

Yes, there’s no substantive disagreement here, really. Rather - at least on my part - there is a “niggle” which essentially boils down to this: does the very concept of a (formal) “semantics” really presuppose something like a set theory? Put differently: does the very notion of “satisfaction” presuppose some kind of set theory (which allows us to make sense of the question “Does X satisfyϕ\phi?”)? Perhaps. And it is certainly the best way to formalize “satisfaction” that is currently available/known to us. But, and this is part of the point I’m making, it is unclear (at least to me) that a formal theory of “satisfaction” must be set-theoretic. (Of course the term “set-theoretic” is sufficiently vague here so as to make this last sentence difficult to assess. If by “set-theoretic” we mean any theory of “collections”, however naive, then perhaps one could say that any formal theory of “satisfaction” will involve collections of some sort - i.e. the collection of all those things that are candidates to satisfy some property - and thereby be set-theoretic in nature.)

Now, in the specific situation that Mike (and this comment thread) is concerned with of course the categoricity results only make sense with a background set theory which, indeed, is what allows us to even express these results as claims to begin with. But this dependence of the semantic properties of second order logic (SOL) on set theory is, I’m arguing, only a “criticism” of second-order-logic with a set theoretic metatheory and not, as it were, “free-floating” SOL. It seems to me that there is nothing intrinsic in “free-floating” SOL that requires it to have a set-theoretic metatheory. It is conceivable that we can give SOL some exotic non-set-theoretic semantics - even though I have no idea what they would look like. But, in any case, it is conceivable. So I think it’s too strong to say that any semantics we give to SOL will always be set-theoretic in nature. Or, to phrase the point in Mike’s terms: it seems too strong a claim to say that any theory of predicates will be a “set theory of sort”.

This has perhaps digressed too far from the main point of this thread, and apologies for this. Just to be clear: what is certainly true - and this, I take it, was Mike’s original point - is that to invoke the strong categoricity properties of SOL as some kind of advantage of SOL over FOL is to forget that these categoricity properties are only provable (indeed expressible) within a (first-order) set theory acting as our metatheory and providing our semantics.

Re: Uncountably Categorical Theories

Obviously there are semantics that are not literally “set-theoretic” in the sense that predicates are literally interpreted by subsets, e.g. you can interpret second-order logic in an elementary topos with predicates interpreted as elements of power objects. But you have to interpret the predicates by something, and it seems likely to me that in order for your semantics to be at least vaguely sound, those somethings will have to behave somewhat like sets.